Early works studying the impact of predictions include [1]} where the algorithm has access to the first coordinate of the cost vector; and [2]} which considered linear costs \(c_t\!=\!\nabla f_t(x_t)\)  and predictions \(\tilde{c}_t\)  with guaranteed correlation \(\tilde{c}_t^\top c_t\ge \alpha \Vert c_t\Vert ^2\) . These predictions improve the regret from \(\mathcal {O}(\sqrt{T})\)  to \(\mathcal {O}(\log T)\) . [3]} considered the case when at most \(B\)  of the predictions fail the correlation condition and provided an \(\mathcal {O}\big ( (1\!+\!\sqrt{B})/\alpha )\log (1\!+T\!-B) \big )\)  regret algorithm, that was further extended to combine multiple predictors [4]}. However, these prior models assume \(\mathcal {X}\)  is time-invariant. A different line of works [5]}, [6]} use adaptive regularizers and define prediction errors \(\varepsilon _t=\Vert c_t - \tilde{c}_t\Vert \)  to obtain \(\mathcal {O}\big (\sqrt{\sum _{t} \varepsilon _t^2}\big )\)  regret bounds. We adapt these methods to the time-varying constrained setting (\(x_t\!\in \!\mathcal {X}_T\) ) where we incorporate predictions for the cost and constraint vectors. Finally, [7]}, [8]}, [9]} consider a fixed \(\mathcal {X}\)  and multiple predictions over a time window, and bound the expected performance or assume some special cost structure.
